The present invention is directed to swinging bob toys with a sliding middle bob having a low moment of inertia about an axis perpendicular to the bore axis of the middle bob, and more particularly to swinging bob toys with a sliding middle bob having a low transient moment of inertia about an axis perpendicular to the bore axis of the middle bob due to movable components within the middle bob.
As shown in FIG. 1A, a swinging bob toy (200) consists of three bobs (210), (211) and (212) on a string (220), with the end bobs (210) and (212) constrained at the ends (221) and (222) of the string (220), and the middle bob (211) having a bore (230) through which the string (220) passes, thereby allowing the middle bob (211) to slide along the string (220). The end bobs (210) and (212) are fixed on the string (220) at the ends (221) and (222) thereof by pins (205) and (not visible in FIG. 1A) lodged into the bores of the end bobs (210) and (212), respectively. (Alternatively, the bobs (210) and (212) may be constrained on the string (220) by knots at each end (221) and (222) of the string (220) having diameters larger than the bores of the end bobs (210) and (212), respectively.) As shown in FIG. 2A, the toy (200) is operated by holding an end bob (212), and oscillating the hand (141) to cause the other two bobs (210) and (211) to separate and the end bob (210) to orbit about the middle bob (211). The bobs (210) and (211) can describe a vertical orbit (290), as shown in FIG. 2A, or horizontal orbits, figure-eight type orbits or irregular paths. Having a bob (210)/(212) at each end (221)/(222) of the string (220) allows a player to hold either end bob (210)/(212) during operation and perform juggling tricks, such as switching end bobs (210)/(212) in mid-air.
As discussed in U.S. Pat. No. Re. 34,208 (column 3, lines 32-57), high-speed photography shows that for a swinging bob toy (200) with a middle bob (211) having a low moment of inertia, the rotation of the middle bob (211) has two different modes of motion as the end bob (210) passes by the string (220) at the top of its orbit, i.e., when the end bob (210) performs its “string pass.”
In a first mode of motion, the bore axis (235) of the middle bob (211) rotates to roughly follow the path of the swinging end bob (210) as it (210) describes the lower half (292) of its orbit (290), as is indicated by the clockwise arrow next to the middle bob (211) in FIG. 4A. But as the swinging end bob (210) begins the upper half (291) of its orbit (290), the rotation of the middle bob (211) slows and stops, as indicated by the lack of an arrow next to the middle bob (211) in FIG. 4B. Then, during the upper half (291) of the orbit (290) of the swinging end bob (210), the middle bob (211) reverses its direction of rotation, as is indicated by the counter-clockwise arrow next to the middle bob (211) in FIG. 4C. As the swinging end bob (210) continues its descent, the middle bob (211) completes a roughly 180° rotation (which according to the lexography of the present specification will be termed the 180° string pass rotation), and the bore axis (235) is roughly horizontal and points towards the side of the orbit (290) where the swinging end bob (210) is currently descending, as is shown in FIG. 4D.
In a second mode of motion, the bore axis (235) of the middle bob (211) rotates to roughly follows the path of the swinging end bob (210) as it (210) describes the lower half (292) of its orbit (290), as is indicated by the clockwise arrow next to the middle bob (211) in FIG. SA. As the swinging end bob (210) begins the upper half (291) of its orbit (290), the rotation of the middle bob (211) slows and stops, as indicated by the lack of an arrow next to the middle bob (211) in FIG. 5B. Then, during the upper half (291) of the orbit (290) of the swinging end bob (210), the middle bob (211) rotates in the horizontal plane to the side of the string (220) on which the outer bob (210) will pass, as is indicated by the arrow next to the middle bob (211) in FIG. 5C. As the swinging end bob (210) continues its descent, the middle bob (211) completes a roughly 180° rotation in the horizontal plane (i.e., its 180° string pass rotation), and again the bore axis (235) is roughly horizontal and points towards the side of the orbit (290) where the swinging end bob (210) is currently descending, as is shown in FIG. 5D.
Hybrid motions of the middle bob (211), combining or alternating between the first and second modes of motion, are also possible. For instance, in the course of its (211) rotation during the string pass, the middle bob (211) may begin to rotate counter-clockwise in the vertical plane, then rotate in the horizontal plane, and then rotate counter-clockwise again in the vertical plane. Or the middle bob (211) may rotate around an axis that is mid-way between the vertical and horizontal planes.
As shown in the cut-away view of the middle bob (211) of FIG. 3A and the cross-sectional view of FIG. 3B, one of the innovations of the swinging bob toy (200) of U.S. Pat. No. Re. 34,208 is a high-density weight (240) centered within a low-density surrounding material (250). In a swinging bob toy marketed under the trademark AstroJax®, and having been distributed by New Toy Classics of San Francisco, Calif., United States, the weight (240) is made of brass and is essentially cylindrical with a central bore (232) along the axis of cylindrical symmetry (235) (i.e., the “polar axis”) of the bob (211). The material (250) surrounding the weight (240) is a soft foam having a density of roughly 0.4 g/cc. The exterior surface (251) of the foam bob (211) is spherical, with the exception of two conical-section indents (231) at the top and bottom which lead to the weight (240). The bore (230) of the bob (211) consists of the conical indents (231) in combination with the bore (232) of the weight (240). The mouth (234) of each conical-section indent (231) is rounded to meet the outside spherical surface (251).
The function of the high-density weight (240) is to concentrate the mass near the center of the bob (211), providing a low moment of inertia I about axes perpendicular to the polar axis (235), thereby allowing the middle bob (211) to rotate rapidly as the swinging outer bob (210) traverses the top (291) of its orbit (290). This is the same principle that a diver uses when she tucks into a ball during a dive to complete more rotations, or an ice skater uses when he brings in his arms during a spin to rotate faster.
The moment of inertia I of a middle bob (211) about an axis of rotation (299) in the equatorial plane (237) is given byI=∫ρr2dτ,  (1.1)where ρ is density, r is distance from the axis of rotation (299), dτ is an infinitesimal volume element, and the integration is performed over volume. (The “moment of inertia” according to the lexography of the present invention is sometime referred to in other literature as the “radius of gyration.”) The dependence of the moment of inertia I on the second power of the distance r from the axis of rotation (299) is somewhat non-intuitive since non-rotational dynamics does not have any relevant quantities with a similar radius-squared weighting. For instance, for a homogeneous ball of radius R, the inner half contributes about 3% to the total moment of inertia, while the outer half contributes about 97% to the total moment of inertia.
As discussed in U.S. Pat. No. Re. 34,208, a crucial measure of the goodness of operation of a swinging bob toy (200) is the dimensionless operation ratio X given byX=(m h2/I)/1/2  (1.2)where m is the mass of a bob (210)/(211)/(212), and h is the height of the bore (230). If the operation ratio X is much greater than unity, the middle bob (211) can rotate rapidly in response to torques produced by the string (220), and so the string (220) will not snag around the middle bob (211) during the string pass and the motion will be smooth. However, if the operation ratio X is much less than unity, the middle bob (211) cannot rotate rapidly in response to torques produced by the string (220), and so the string (220) will tend to snag, or even tangle, around the middle bob (211) during the string pass, disrupting the orbital motions of the bobs (210) and (211) and inhibiting enjoyment of the toy (200).
Active People of Benningen, Switzerland and a growing number of other toy companies are producing swinging bob toys which utilize a metal weight to lower the moment of inertia. Unfortunately, the limited ranges in the densities of solid low-density and high-density materials limits the degree to which the operation ratio X can be maximized, and ways in which the middle bob may be constructed. Furthermore, the inclusion of a metal weight contributes substantially to the cost of the toy.
It is an object of the present invention is to provide a swinging bob toy which operates smoothly, i.e., a swinging bob toy where the string does not tend to tangle around the middle bob, and the string tension does not have spikes, jumps, or vary rapidly.
It is another object of the present invention to provide a swinging bob toy having a middle bob with a small moment of inertia and a large operation ratio.
It is another object of the present invention is to provide a swinging bob toy having a middle bob which does not incorporate a high-density, centrally-positioned weight yet still has a small moment of inertia and a large operation ratio.
It is another object of the present invention is to provide a swinging bob toy with a dynamic moment of inertia, i.e., a moment of inertia which is time-dependent, velocity-dependent, acceleration-dependent, or dependent on a history of the motion of the middle bob.
It is another object of the present invention is to provide a swinging bob toy with a dynamic operation ratio, i.e., a goodness of operation ratio which is time-dependent, velocity-dependent, acceleration-dependent, or dependent on a history of the motion of the middle bob.
It is another object of the present invention is to provide a swinging bob toy with a middle bob having movable components to produce a small moment of inertia and a large operation ratio.
It is another object of the present invention is to provide a swinging bob toy with a middle bob having a liquid-containing bladder to produce a small moment of inertia and a large operation ratio.
It is another object of the present invention is to provide a swinging bob toy with a middle bob having a bladder containing one or more liquids, where the density and viscosity of the liquid and the geometry of the bladder produce a small moment of inertia and a large operation ratio.
Furthermore, as motivated and explained in detail below, it is an object of the present invention to provide a swinging bob toy having a middle bob with a liquid-containing bladder where the liquid has a viscosity which is large enough that the operation is smooth as the orbiting outer bob enters the bottom of its orbit.
Additional objects and advantages of the invention will be set forth in the description which follows, and will be apparent from the description or may be learned from the practice of the invention. The objects and advantages of the invention may be realized and obtained by means of the instrumentalities and combinations particularly pointed out in the claims.